Trading the equal-weight tilt more slowly improved its information ratio, and the gain appears larger than the costs saved.
Ceco, Shi and Wong treat the position as a benchmark-relative bet: hold the index, then size an equal-weight tilt by active risk. The expected return comes from harvesting dispersion (volatility harvesting) and timing concentration. Over 1995-2024, they report cumulative net wealth above both the cap-weighted and equal-weighted S&P 500 benchmarks, with a higher information ratio than equal weight.
Why concentration matters
The starting point is an identity from stochastic portfolio theory. Market diversity φ is the average log market weight across the index, or the log of its geometric mean weight. For the S&P 500, it has averaged about -6.95. Equal weight's log return relative to cap weight splits into the change in φ and half the accumulated dispersion δ. Dispersion is the excess growth earned by rebalancing among stocks as their weights move relative to one another.
Dispersion contributes positively. Diversity falls when a few mega caps run ahead, and that fall can overwhelm the rebalancing gain. In the paper's chart, diversity now stands at a level last seen around the dot-com bubble. Forecasting equal weight against the index therefore requires a view on both diversity and dispersion.
The authors give each a process. Under mean reversion, diversity moves toward a long-run level, with volatility proportional to the square root of dispersion. Log dispersion follows an Ornstein-Uhlenbeck process. Their trending alternative uses an exponentially smoothed trend in daily diversity changes (21-day half-life) and GARCH(1,1) variance.
The trade is a tilt λ on a long-short portfolio that buys equal weight and sells the market. At 0, the investor holds the index; at 1, equal weight. A negative tilt shorts equal weight against the index. With no frictions, the target responds immediately to the forecast. It takes expected relative drift, diversity drift plus half of dispersion, divides by active variance, adds a constant half for the mixing term, and scales the result by 1/(1+γ).
Slowing the trade
A changing forecast makes that frictionless target jump. The authors impose quadratic penalties on the size of the tilt, Λ1, and its rate of change, Λ2. The first stands in for the constituent rebalancing required to hold equal weight. Their linear forward-backward stochastic differential equation produces a rule following Gârleanu and Pedersen's "aiming in front of a moving target" logic: trade toward a discounted average of future forecast targets. Each month, the new tilt blends last month's position with that aim. The mean-reverting model looks 12 months ahead.
These penalties stand in for costs; they are not the costs deducted from reported wealth. The authors calibrate them on 1977-1994 to match the cumulative wealth drag from 15bp proportional costs, then deduct the actual 15bp in the return series. For the mean-reverting model, calibration gives Λ1 = 1.17 and Λ2 = 0.54. They choose γ = 19.91 for 5% in-sample active volatility.
The data cover CRSP's changing S&P 500 membership from 1977 to 2024. To estimate dispersion, the authors multiply monthly realized equal-weight dispersion by 0.21. That fitted scaling has an adjusted R² of 0.95 and accounts for leakage, monthly rebalancing and proxy error; it implies that about a fifth of raw dispersion survives in relative return. Parameters are frozen at 1994. Over the subsequent 360 months, 1995-2024, the mean-reverting frictional book reaches cumulative net wealth of 44.92, against 24.33 for equal weight and 21.34 for the market. Its index-relative information ratio is 0.35. The trending model posts 0.29 and plain equal weight 0.13.
Did it earn its costs?
The mean-reverting policy's turnover falls from 0.41 to 0.15 when the trading penalty is applied, while its net IR rises from 0.21 to 0.35. Gross Sharpe rises too, from 0.69 to 0.74. Saved transaction costs cannot account for a gross improvement. The authors say the gain "cannot be entirely explained by a reduction in transaction costs," and suggest that the slower signal better forecasts slow reversion. Their description of the change across both models as "slightly improving," suits the trending result, whose IR moves from 0.26 to 0.29 as turnover falls from 0.59 to 0.34. It sells the mean-reverting result short. The next sentences call that gain a "significant improvement in relative performance."
The simulation gives a different view of gross performance. Across the paper's 10,000 simulated 48-year paths, smoothing wins after surrogate penalties and trails slightly before them. On average across those 10,000 paths, it gives up gross return before surrogate penalties. The historical path instead shows gross Sharpe climbing from 0.69 to 0.74. That supports the authors' interpretation, though it is one history.
The long-only result deserves attention. Restricting λ to [0, 1] preserves a mean-reverting IR of 0.34 with turnover of 0.05. The unrestricted book averages gross exposure of 1.46, and we did not find a short-borrow charge beyond the 15bp. For the IR, the clipped result makes that omission mostly moot. Wealth tells a different story: the positions outside [0, 1], including shorts and leverage, account for much of the gap between 32.46 and 44.92.
The advantage is relative to the index. Net Sharpes are 0.72 for the strategy, 0.61 for the market and 0.58 for equal weight. The paper calls those differences immaterial. Since the objective is benchmark-relative performance, IR is the appropriate measure, though 0.35 over 360 months works out to roughly a t of 1.9. The Fama-French three-factor regression gives alpha of 2.47% a year (t = 1.943) and a value (HML) loading of 0.254 (t = 8.15). With five factors, alpha drops to 1.54% (t = 1.18). The authors describe these results as "descriptive factor attribution rather than conclusive evidence of persistent abnormal returns," and I agree. Plain equal weight has a -0.41% three-factor alpha, leaving size and value exposure to explain its premium in this sample.
Across a broad range of Λ2 settings, the mean-reverting IR remains fairly steady. Large Λ2 values, however, push the trending IR below zero. The authors chose the 12-month forecast horizon because they "find this reasonably balances" costs and forecast noise. We could not tell from the text which sample informed that choice.
What if diversity keeps falling?
Diversity increments look Gaussian, and the model fits them. Dispersion strains the specification. Fitted standard deviation of log dispersion rises from 0.71 over 1977-2004 to 1.46 over 2005-2024, driven in large part, the authors say, by 2008-09 and 2020. They acknowledge that its "multi-scale and nonstationary behaviour" cannot be faithfully captured by an Ornstein-Uhlenbeck process.
The fitted parameters depend on the window as well. Dispersion reversion speed moves from 5.52 on 1977-1994 to 3.179 on 1977-2004, while dispersion volatility moves from 0.99 to 2.369. The authors also flag a shift in diversity volatility. They attribute the changes partly to elevated dot-com dispersion, which appears only in the longer estimation sample.
The backtest fixes long-run diversity at -6.93 and reversion speed at 0.43, implying a half-life near 1.6 years. The paper reports recent relative underperformance as diversity remains below its mean. Its authors leave open whether the decline is unsustainable, as it was before the dot-com crash, or whether reversion has broken. That precedent gives us one episode. A target estimated on 1977-1994 and left unchanged will continue leaning toward equal weight while concentration persists. A recovery in diversity that repays that frozen model would change my mind.
Our top-500 adaptation
We could not trade historical S&P 500 membership. Instead, our adaptation uses an annually selected top-500 US non-ADR universe. We ran the mean-reverting frictional rule monthly from January 2020 to July 2024, filled at the next close, and charged $0.004 a share with a $1 minimum per order on a $1,000 start. These figures come from our backtest: 0.03% total return, 0.24 Sharpe and -0.04% maximum drawdown across 27,170 trades.
The paper's mean-reverting book reports 14% annualized net return, a 0.72 net Sharpe and 16% annualized volatility over 1995-2024. Our result is much lower. Different universes, windows and cost models limit what that gap says about the paper.
Our own book had volatility of 0.03% and beta of 0.00 to SPY. It behaved like cash, whereas the paper's book has a market loading of 0.981. We have not established why ours remained uninvested. Our run also omitted the paper's procedures for setting γ, Λ1 and Λ2, and its window falls within the recent period when the paper reports the strategy lagging. On such a small book, per-order minimums likely cost far more than 15bp per trade. Changing the universe could move diversity and dispersion inputs either way. We cannot fully explain the gap from what we can see. Our run produced no information ratio, leaving the paper's active edge untested by this automated pass rather than judged by it.
The paper's case rests on the smoothed tilt's 0.35 IR and the clipped version's 0.34. The current concentration regime remains the test it has yet to pass.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.